The problem seems to be coming from some other gadget that's getting stuck and needs to be jiggled afresh with each flush. I don't recognize the thing but I suspect it's some kind of water-saving device. I also suspect that it's required by law and that I will face stiff fines if I tamper with it; but I don't know for sure. In any case I will probably end up calling a plumber so that I don't find myself, so to speak, in the shit.
The brand name stamped on the hardware proclaims "AMERICAN STANDARD - MADE IN MEXICO."
And such is life in the 21st century. Anyway, it's a beautiful day out, and I may just get out for a little fresh air. It's been hot and muggy the past few days and this is a welcome break.
BREAKING: It's fixed! Closer inspection revealed that there's a threaded piece that adjusts the position of the lever relative to the float, so you can make the float move the lever more. Problem is now resolved.
You may carry on.